= Codimension-one Stiefel–Whitney immersion obstruction
{title2=$\overline w_2(TM)=w_1(TM)^2+w_2(TM)=0$}
An <immersion> of a smooth $d$-manifold in $\mathbb R^{d+1}$ supplies a rank-one <normal bundle> $\nu$ with $TM\oplus\nu\cong\varepsilon^{d+1}$. The <Whitney product formula for Stiefel–Whitney classes> therefore requires $w(\nu)=w(TM)^{-1}$. A <real line bundle> has no positive-degree classes beyond $w_1$, so every higher-degree part of the inverse total class must vanish. For $\mathbb{RP}^2\times\mathbb{RP}^2$, the <stabilized tangent bundle of real projective space> gives $w(TM)^{-1}=(1+a)(1+b)$ in $\mathbb F_2[a,b]/(a^3,b^3)$. Its nonzero degree-two part $ab$ excludes an <immersion> in $\mathbb R^5$ even though the normal line need not be orientable.
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